Past exam of the quantum information course of the University of Cambridge / 2016 / Paper 1 / 3

A weak measurement couples an observable only weakly to a meter, so the pointer distributions associated with its different eigenvalues overlap strongly. One trial supplies little information and a small disturbance; an ensemble is needed to estimate a small mean shift. It is not an exact determination of a previously hidden eigenvalue with no disturbance.

For pointer position $q$ and conjugate momentum $p$, a short von Neumann measurement interaction has unitary $U=\exp (-igA\otimes p/\hslash )$. An eigenvalue $a$ translates the pointer by $ga$. The weak regime requires those translations to be small relative to the pointer width. Without postselection the first-order mean displacement is $g⟨A⟩$ for an initially uncorrelated, suitably centered meter.

If the system is prepared in $|i⟩$ and postselected in $|f⟩$, the unnormalized meter state isFor a real centered Gaussian pointer with no position–momentum correlation, the conditional position shift is $g\mathrm{Re}{A}_w$; the imaginary part affects the conjugate momentum distribution. The weak value can be complex or have a real part outside the observable's eigenvalue interval, because it is a conditional transition-amplitude ratio, not an eigenvalue.

An example apparatus uses photon polarization as the two-level system, a weak polarization-dependent transverse displacement as the interaction, and an output polarizer for postselection. Repeated position detections estimate the selected beam's displacement. Nearly orthogonal selection can make the conditional shift large, but also reduces the accepted sample and eventually invalidates the first-order expansion. The original weak-value proposal emphasized preselection, a weak coupling and postselection, rather than an anomalous result from an ordinary sharp spin measurement.^[ref]

Weak coupling limits quantum measurement backaction per trial; it does not abolish the information–disturbance tradeoff. Repeating weak observations can accumulate substantial backaction, and discarding rejected trials cannot automatically improve the information available per prepared system. Conditional pointer amplification and a statistically improved parameter estimate are distinct claims.

Quantum teleportation transfers an unknown qubit state using a previously shared Bell pair, local operations and two classical bits. It requires no description of the unknown complex amplitudes and produces no extra copy.

Let the input be $|\chi {⟩}_S=\alpha |0⟩+\beta |1⟩$, and let Alice and Bob share $|{\Phi }^{+}{⟩}_{AB}$. Alice applies a controlled-NOT gate from $S$ to $A$, then a Hadamard gate to $S$. Expanding the circuit gives the Bell-basis teleportation identity in computational-output form:Her projective measurement yields each bit pair $(a,b)$ with probability $1/4$. She sends the two classical bits, and Bob applies $Z^a{X}^b$ to recover $|\chi ⟩$. The order is the inverse of the displayed conditional operator; interchanging these Pauli corrections changes only an irrelevant global phase when both bits are one.

Before receiving the bits, Bob hasThus teleportation respects the no-communication theorem and cannot transmit the usable unknown state faster than the classical message. Alice's input has been measured, and the shared entanglement resource has been consumed, in agreement with the no-cloning theorem. The protocol transfers the state, not the material particle initially carrying it.

A physical implementation must distribute and preserve an entangled pair, implement a Bell-basis measurement and transmit its result to a controllable output system. Imperfect entanglement, lost outcomes and noisy corrections reduce quantum fidelity; a heralded probabilistic experiment must report its success probability separately from conditional fidelity. Ideal deterministic teleportation assumes every Bell outcome can be identified and corrected.

Quantum error correction encodes a logical qubit into a larger Hilbert space, diagnoses a specified error family with an error syndrome, and reverses it without learning the protected logical amplitudes. It must preserve arbitrary superpositions, not just distinguish two classical codewords.

The bit-flip repetition code encodesMeasure the commuting stabilizer generators $Z_1{Z}_2$ and $Z_2{Z}_3$ using ancillary qubits. Their outcomes reveal the parity relations, with the three-qubit bit-flip syndrome tableBoth codewords have the same parity checks, so the syndrome does not reveal $\alpha$ or $\beta$. For example, the $X_2$ error produces $\alpha |010⟩+\beta |101⟩$; applying $X_2$ restores the encoded superposition. A coherent combination of correctable errors is also recoverable after the syndrome is recorded in an ancilla, by linearity.

This three-qubit code corrects one bit flip, not every one-qubit error. A physical $Z_1$ changes the relative logical phase without a nontrivial bit-flip syndrome. General single-qubit noise is a linear combination of $I,X,Y,Z$; protecting arbitrary single-qubit errors therefore needs a code such as the Shor code, the Steane code or the five-qubit code, with the corresponding enlarged syndrome structure.

The general exact-recovery criterion is the Knill--Laflamme condition. For code projector $P$ and error operators $E_r$, there must be constants $C_{rs}$ such thatIt expresses that the syndrome information can depend on the error but not on the logical state. Encoding entangles several physical qubits; it does not create independent copies of an unknown input, so it respects the no-cloning theorem. Fault-tolerant implementations must also prevent noisy syndrome extraction and recovery gates from spreading an initially correctable error into an uncorrectable one.